Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 4J (Statistical Modelling)

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- Solution 4J

Solution

The standard normal distribution function

Φ

gives

P (Z_{i} = 1 ∣ x_{i}) = P (ϵ_{i} > lo g c - μ - x_{i}^{T} β) = Φ (\frac{μ - l o g c}{σ} + x_{i}^{T} \frac{β}{σ}) .

The observations are independent Bernoulli random variables, because the original errors are independent. Thus the probit regression has intercept

α = (μ - lo g c) / σ

and coefficient vector

γ = β / σ

. With the specified known

σ

and

c

, recovery is

μ = lo g c + σ α, β = σγ .

The scale would not be recoverable from threshold data alone if

σ

were unknown.

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